An optimal lower bound for distinct elements in the message passing model

In the message-passing model of communication, there are k players each with their own private input, who try to compute or approximate a function of their inputs by sending messages to one another over private channels. We consider the setting in which each player holds a subset Si of elements of a universe of size n, and their goal is to output a (1 + ε)-approximation to the total number of distinct elements in the union of the sets Si with constant probability, which can be amplified by independent repetition. This problem has applications in data mining, sensor networks, and network monitoring. We resolve the communication complexity of this problem up to a constant factor, for all settings of n, k and ε, by showing a lower bound of Ω(k · min(n, 1/ε2) + k log n) bits. This improves upon previous results, which either had non-trivial restrictions on the relationships between the values of n, k, and ε, or were suboptimal by logarithmic factors, or both.